This project in random matrix theory stems from Dyson diffusion for the eigenvalues of a random matrix ,which forces the eigenvalues to evolve according to non-intersecting Brownian motion.Upon letting the size of the random matrices grow arbitrarily large, the eigenvalues turn into a "Markov cloud " of infinite non-intersecting particles,distributed according to a certain equilibrium measure. For each given time,its support will be concentrated on intervals, whose number may vary with time..Therefore, when time evolves, intervals may merge, may disappear and be created, leading to a region R in space- time,whose boundary will be regular,except for various singularities. Near the boundary points of R the non-intersecting Brownian motions will, in the limit, tend to a Markov cloud performing phase transitions when approaching a singularity; these infinte dimensional diffusion are thus critical phenomena and should exhibit universal properties.We wish to derive (nonlinear) PDE's for the transition probabilities and various scaling limits which will yield boundary conditions, appropriatedly understood ,for these Painleve type PDE"s.This will also be a tool to pass from one critical phenomena to another.Along the same vein, another goal of the project is to connect conformal maps, dispersionless 2D-Toda and the Stochastic Lowner equation, through using a stochastically changing domain, via Brownian motion.
Random matrix theory has a diverse interface with numerous mathematical and physical disciplines, on the one hand, Fredholm determinants ,integrable mechanics and Painleve equations and on the one hand conformal field theory and statistical mechanics, and in particular. critical phenomena and universality.The basic motivation is to tie these topics together using a Painleve type theory of partial differential equations to explain how various critical phenomena merge into each other and emerge out of each other, perphaps creating a sort of familty-tree for various critical phenomena.The tools of the various fields alluded to will come into play in both describing the phenomena and deriving equations for the probabilistic prediction of how the phenomena evolves.
